Rolling of manifolds without spinning

The control model of rolling of a Riemannian manifold (M; g) onto another one $ \left( {\hat{M},\hat{g}} \right) $ consists of a state space Q of relative orientations (isometric linear maps) between their tangent spaces equipped with a so-called rolling distribution $ {\mathcal D} $ _R, which models the natural constraints of no-spinning and no-slipping of the rolling motion. It turns out that the distribution $ {\mathcal D} $ _R can be built as a sub-distribution of a so-called no-spinning distribution $ {{\mathcal{D}}_{\overline{\nabla}}} $ on Q that models only the no-spinning constraint of the rolling motion. One is thus motivated to study the control problem associated to $ {{\mathcal{D}}_{\overline{\nabla}}} $ and, in particular, the geometry of $ {{\mathcal{D}}_{\overline{\nabla}}} $ -orbits. Moreover, the definition of $ {{\mathcal{D}}_{\overline{\nabla}}} $ (contrary to the definition of $ {\mathcal D} $ _R) makes sense in the general context of vector bundles equipped with linear connections. The purpose of this paper is to study the distribution $ {{\mathcal{D}}_{\overline{\nabla}}} $ determined by the product connection $ \nabla \times \hat{\nabla} $ on a tensor bundle $ {E^{*}}\otimes \hat{E}\to M\times \hat{M} $ induced by linear connections ?, $ \hat{\nabla} $ on vector bundles $ E\to M,\,\,\,\hat{E}\to \hat{M} $ . We describe completely the orbit structure of $ {{\mathcal{D}}_{\overline{\nabla}}} $ in terms of the holonomy groups of ?, $ \hat{\nabla} $ and characterize the integral manifolds of it. Moreover, we describe the general formulas for the Lie brackets of vector elds in $ {E^{*}}\otimes \hat{E} $ in terms of $ {{\mathcal{D}}_{\overline{\nabla}}} $ and the vertical tangent distribution of $ {E^{*}}\otimes \hat{E}\to M\times \hat{M} $ . In the particular case of tangent bundles $ TM\to M,\,\,\,T\hat{M}\to \hat{M} $ and Levi-Civita connections, we describe in more detail how $ {{\mathcal{D}}_{\overline{\nabla}}} $ is related to the above mentioned rolling model, where these Lie brackets formulas provide an important tool for the study of controllability of the related control system.     

Solutions of singular control systems on lie groups

Let G be a connected Lie group with Lie algebra $ \mathfrak{g} $ . A singular control system $ {\mathcal{S}_G} $ on G is defined by a pair (E, D) of $ \mathfrak{g} $ -derivations. Through a fiber bundle decomposition of TG in [1] ; the authors decompose $ {\mathcal{S}_G} $ in two subsystems $ {\mathcal{S}_{G/V}} $ and $ {\mathcal{S}_V} $ ; as in the linear case on Euclidean spaces, see for instance [9] : Here, $ V \subset G $ is the Lie subgroup with Lie algebra $ \mathfrak{v} $ ; the generalized 0-eigenspace of E: On the other hand, D defines the drift vector field of the system. We assume that the subspace $ \mathfrak{v} $ is invariant under D. With this hypothesis we show a process to determine the solution of $ {\mathcal{S}_G} $ through every state x?=?yv; where v is any admissible initial condition on V. From this information, we are able to build the global solution. Finally, in order to illustrate our processes we develop some examples on nilpotent simply connected Lie groups.     

Monotonic homotopy for trajectories of young systems

Let $ \mathbb{Y} $ be a Young system. Assume that the accessible set $ \mathcal{A} $ ( $ \mathbb{Y} $ ; x) of $ \mathbb{Y} $ starting from x is locally and semi-locally simply connected by trajectories of $ \mathbb{Y} $ . We prove that the covering space Γ( $ \mathbb{Y} $ ; x) of p-monotonically homotopic trajectories is identified to the universal covering space of $ \mathcal{A} $ ( $ \mathbb{Y} $ ; x).     

Explicit solutions of the {\mathfrak{a}_1} -type lie-Scheffers system and a general Riccati equation

For a general differential system $ \dot{x}(t) = \sum\nolimits_{d = 1}^3 {u_d } (t){X_d} $ , where X _d generates the simple Lie algebra of type $ {\mathfrak{a}_1} $ , we compute the explicit solution in terms of iterated integrals of products of u _d ’s. As a byproduct we obtain the solution of a general Riccati equation by infinite quadratures.     

Controllability of control systems on complex simple lie groups and the topology of flag manifolds

Let S be a subsemigroup with nonempty interior of a connected complex simple Lie group G. It is proved that S = G if S contains a subgroup G(α) ≈ Sl (2, $ \mathbb{C} $ ) generated by the exp $ {{\mathfrak{g}}_{{\pm \alpha }}} $ , where $ {{\mathfrak{g}}_{\alpha }} $ is the root space of the root α. The proof uses the fact, proved before, that the invariant control set of S is contractible in some flag manifold if S is proper, and exploits the fact that several orbits of G(α) are 2-spheres not null homotopic. The result is applied to revisit a controllability theorem and get some improvements.     

Controllability of the Semilinear Beam Equation

In this paper, we prove the approximate controllability of the following semilinear beam equation: $$ \left\{ \begin{array}{lll} \displaystyle{\partial^{2} y(t,x) \over \partial t^{2}} & = & 2\beta\Delta\displaystyle\frac{\partial y(t,x)}{\partial t}- \Delta^{2}y(t,x)+ u(t,x) + f(t,y,y_{t},u),\; \mbox{in}\; (0,\tau)\times\Omega, \\ y(t,x) & = & \Delta y(t,x)= 0 , \ \ \mbox{on}\; (0,\tau)\times\partial\Omega, \\ y(0,x) & = & y_{0}(x), \ \ y_{t}(x)=v_{0}(x), x \in \Omega, \end{array} \right. $$ in the states space $Z_{1}=D(\Delta)\times L^{2}(\Omega)$ with the graph norm, where β?>?1, Ω is a sufficiently regular bounded domain in IR ^N , the distributed control u belongs to L ²([0,τ];U) (U?=?L ²(Ω)), and the nonlinear function $f:[0,\tau]\times I\!\!R\times I\!\!R\times I\!\!R\longrightarrow I\!\!R$ is smooth enough and there are a,c?∈?IR such that $a<\lambda_{1}^{2}$ and $$ \displaystyle\sup\limits_{(t,y,v,u)\in Q_{\tau}}\mid f(t,y,v,u) - ay -cu\mid<\infty, $$ where Q _τ ?=?[0,τ]×IR×IR×IR. We prove that for all τ?>?0, this system is approximately controllable on [0,τ].     

Invariant Cones for Semigroups in Transitive Lie Groups

Let G be a connected Lie group which has a transitive representation on $ \mathbb{R}^{ n } $ and S ? G be a semigroup with a nonempty interior. We study necessary and sufficient conditions for the existence of the S-invariant pointed and generating cone W ? $ \mathbb{R}^{ n } $ , where the parabolic type of S arises as a central concept.     

Null controllability of degenerate/singular parabolic equations

The purpose of this paper is to provide a full analysis of the null controllability problem for the one dimensional degenerate/singular parabolic equation $ {u_t} - {\left( {a(x){u_x}} \right)_x} - \frac{\lambda }{{{x^\beta }}}u = 0 $ , (t, x) ∈ (0, T) × (0, 1), where the diffusion coefficient a(?) is degenerate at x = 0. Also the boundary conditions are considered to be Dirichlet or Neumann type related to the degeneracy rate of a(?). Under some conditions on the function a(?) and parameters β, λ, we prove global Carleman estimates. The proof is based on an improved Hardy-type inequality.     

Sub-Riemannian Homogeneous Spaces of Engel Type

A sub-Riemannian manifold is a smooth manifold which carries a metric defined only on a smooth distribution $\mbox{$\cal D$}$ . In this paper, we will restrict our attention to sub-Riemannian manifolds where the associated distribution is an Engel distribution which means that $\mbox{$\cal D$}$ is a regular and bracket-generating distribution of codimension 2 in a four-dimensional manifold. We obtain a parallelism on a sub-Riemannian structure of Engel type, and then, we classify all simply connected four-dimensional sub-Riemannian manifolds which are homogeneous spaces by using a canonical linearization of the structure.     

Nonlinear evolution inclusions with one-sided perron right-hand side

In a Banach space X with uniformly convex dual, we study the evolution inclusion of the form $ {x}^{\prime}(t)\in Ax(t)+F\left( {x(t)} \right) $ , where A is an m-dissipative operator and F is an upper hemicontinuous multifunction with nonempty convex and weakly compact values. If X^* is uniformly convex and F is one-sided Perron with sublinear growth, then, we prove a variant of the well known Filippov-Pli? theorem. Afterward, sufficient conditions for near viability and (strong) invariance of a set $ K\subseteq \overline{D(A)} $ are established. As applications, we derive ε - δ lower semicontinuity of the solution map and, consequently, the propagation of continuity of the minimum time function associated with the null controllability problem.     

Study on Existence of Solutions for p-Kirchhoff Elliptic Equation in ℝN with Vanishing Potential

In this paper, we study the existence of positive solutions to p?Kirchhoff elliptic problem \(\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllll} &\left(a+\mu\left({\int}_{\mathbb{R}^{N}}\!(|\nabla u|^{p}+V(x)|u|^{p})dx\right)^{\tau}\right)\left(-{\Delta}_{p}u+V(x)|u|^{p-2}u\right)=f(x,u), \quad \text{in}\; \mathbb{R}^{N}, \\ &u(x)>0, \;\;\text{in}\;\; \mathbb{R}^{N},\;\; u\in \mathcal{D}^{1,p}(\mathbb{R}^{N}), \end{array}\right.\!\!\!\! \\ \end{array} \) ?????(0.1) where a, μ > 0, τ > 0, and f(x, u) = h ₁(x)|u| ^m?2 u + λ h ₂(x)|u| ^r?2 u with the parameter λ ∈ ?, 1 < p < N, 1 < r < m < \(p^{*}=\frac {pN}{N-p}\) , and the functions h ₁ (x), h ₂(x) ∈ C(?^N) satisfy some conditions. The potential V(x) > 0 is continuous in ? ^N and V(x)→0 as |x|→+∞. The nontrivial solution forb Eq. (1.1) will be obtained by the Nehari manifold and fibering maps methods and Mountain Pass Theorem.     

An intrinsic formulation of the problem on rolling manifolds

We present an intrinsic formulation of the kinematic problem of two n-dimensional manifolds rolling one on another without twisting or slipping. We determine the configuration space of the system, which is an n(n?+?3)/2-dimensional manifold. The conditions of no-twisting and no-slipping are encoded by means of a distribution of rank n. We compare the intrinsic point of view versus the extrinsic one. We also show that the kinematic system of rolling the n-dimensional sphere over $ {\mathbb{R}^n} $ is controllable. In contrast with this, we show that in the case of SE(3) rolling over $ \mathfrak{s}\mathfrak{e}(3) $ the system is not controllable, since the configuration space of dimension 27 is foliated by submanifolds of dimension 12.     

Special representations of ℤ n -actions

In this paper we generalize the following statement (Alpern's theorem). Given a relatively prime set $$\{ h_i \} \subset \mathbb{N},i = 1,...,N \leqslant \omega ,$$ , and a probability distribution {α _i }, for any antiperiodicT there is a representation $X = \coprod\nolimits_{i = 1}^N {\left( {\coprod\nolimits_{j = 0}^{h_i - 1} {T^j B_i } } \right)}$ , where μ(B _i )=α _i /h _i . Our main result is the similar statement for free ? ⁿ -actions. Both theorems are generalizations of the well-known Rokhlin-Halmos lemma.     

Homotopy of dynamical systems on manifolds and Morse theory on covering space

We consider a dynamical system X on a compact differentiable manifold M and the induced dynamical system X(ρ) on the universal covering space $ \tilde M $ of M. We develop algebraic topology methods for estimating the lower bounds on the number of codimension one surfaces (i.e. on the number of index one equilibria) on the boundary of regions of stability on $ \tilde M $ . We also develop a method of constructively verifying that the number of index one equilibria on the boundary of any region of stability in $ \tilde M $ is preserved during a homotopy of vector fields, avoiding a verification of the transversality condition. This approach allows us to get lower bounds for the index one equilibria on the boundary of stability regions of dynamical systems on noncompact manifolds and get stronger estimates than the ones afforded by the classical Morse –Smale theory.     

Uniform Stability of the Solution for a Memory-Type Elasticity System with Nonhomogeneous Boundary Control Condition

In this paper, we consider the memory-type elasticity system \(\boldsymbol {u}_{tt}-\upmu {\Delta }{\boldsymbol {u}}-(\upmu +\lambda )\nabla (\text {div}\boldsymbol {u})+{{\int }^{t}_{0}}g(t-\tau ){\Delta }{\boldsymbol {u}}(s)ds=0,\) with nonhomogeneous boundary control condition and establish the uniform stability result of the solution. The exponential decay result and polynomial decay result in some literature are the special cases of this paper.     

Real-Formal Orbital Rigidity for Germs of Real Analytic Vector Fields on the Real Plane

For \(n \geqslant 2\), we consider \(\mathcal {V}^{\mathbb {R}}_{n}\) the class of germs of real analytic vector fields on \(\left (\mathbb {R}^{2}, \widehat {0}\right )\) with zero (n?1)-jet and nonzero n-jet. We prove, for generic germs of \(\mathcal {V}^{\mathbb {R}}_{n}\), that the real-formal orbital equivalence implies the real-analytic orbital equivalence, that is, the real-formal orbital rigidity takes place. This is the real analytic version of Voronin’s formal orbital rigidity theorem.     

Controllability of Right-Invariant Systems on Solvable Lie Groups

We study controllability of right-invariant control systems $\Gamma = A + \mathbb{R}B$ on Lie groups. Necessary and sufficient controllability conditions for Lie groups not coinciding with their derived subgroup are obtained in terms of the root decomposition corresponding to the adjoint operator ad B. As an application, right-invariant systems on metabelian groups and matrix groups, and bilinear systems are considered.     

On High Dimensional Schrödinger Equation with Quasi-Periodic Potentials

In this paper, we consider the high dimensional Schrödinger equation \( -\frac {d^{2}y}{dt^{2}} + u(t)y= Ey, y\in \mathbb {R}^{n}, \) where u(t) is a real analytic quasi-periodic symmetric matrix, \(E= \text {diag}({\lambda _{1}^{2}}, \ldots , {\lambda _{n}^{2}})\) is a diagonal matrix with λ _j >0,j=1,…,n, being regarded as parameters, and prove that if the basic frequencies of u satisfy a Bruno-Rüssmann’s non-resonant condition, then for most of sufficiently large λ _j ,j=1,…,n, there exist n pairs of conjugate quasi-periodic solutions.     

First- and Second-Order Integral Functionals of the Calculus of Variations which Exhibit the Lavrentiev Phenomenon

We study the possible mechanisms of occurrence of the Lavrentiev phenomenon for the basic problem of the calculus of variations $$\mathcal{J}(x) = \int\limits_0^1 {L(t,x(t),\dot x(t))dt \to \inf , x(0) = x_0 } , x(1) = x_1$$ ,when the infimum of the problem in the class of absolutely continuous functionsW _1,1[0, 1] is strictly less than the infimum of the same problem in the class of Lipshitzian functionsW _1,∞[0, 1]. We suggest an approach to constructing new classes of integrands which exhibit the Lavrentiev phenomenon (Theorem 2.1). A similar method is used to construct (Theorem 3.1) a class of autonomousC ¹-differentiable integrandsL(x, .x, ..x) of the calculus of variations which are regular, i.e., convex, coercive w.r.t. ..x, and exhibit theW _2,1–W _2,∞ Lavrentiev gap, i.e., for some choice of boundary conditions of the variational problem $$\begin{array}{*{20}c} {\mathcal{J}(x( \cdot )) = \int\limits_0^1 {L(x(t),\dot x(t),\ddot x(t)) dt \to \inf ,} } \\ {x(0) = x_0 , \dot x(0) = \upsilon _0 , x(1) = x_1 , \dot x(1) = \upsilon _1 } \\ \end{array}$$ ,the infimum of this problem over the spaceW _{2, 1}[0, 1] is strictly less than its infimum over the spaceW _2,∞[0, 1]. This provides a negative answer to the question of whether functionals with regular autonomous second-order integrands should only have minimizers with essentially bounded second derivative.